A triangle has corners A, B, and C located at #(4 ,2 )#, #(3 ,4 )#, and #(6 ,8 )#, respectively. What are the endpoints and length of the altitude going through corner C?

Question

A triangle has corners A, B, and C located at #(4  ,2   )#, #(3  ,4    )#, and #(6 ,8   )#, respectively. What are the endpoints and length of the altitude going through corner C?

Caroline Aucoin · Accepted Answer

Length of altitude (CD) passing through point C #= 4.4721#

Equation of side AB
#(y-2)/(4-2) = (x-4) / (3-4)#
#-1( y-2) = 2 (x-4)#
#2x + y = 10#. Eqn (1)

Let Slope of side AB be ‘m’
#m = (4-2) / (3-4) = -2#
Slope of perpendicular line to AB is #= -(1/m) = -(1/-2) = 1/2#

Eqn of Altitude(CD) to AB passing through point C is
#(y-8) =(1/2)(x-6)#
#2y- 16 = x - 6#

#x - 2y = -10#. Eqn (2)

Solving Eqns (1) & (2) we get the base of the altitude (CD) passing through point C.

Solving the two equations, we get
#x=2, y=6#

Length of the altitude passing through point C
#= sqrt((6-2)^2 + (8-6)^2) = sqrt(16 + 4) = 4.4721#

Length of Altitude (CD) passing through point C = 4.4721#

Noah Atkinson · Answer

undefined To find the altitude going through corner C, we first need to determine the equation of the line containing side AB. Then, we can find the perpendicular line passing through point C, which will be the altitude.

1. Find the equation of line AB:
   - Slope of AB = $ \frac{4 - 2}{3 - 4} = -2 $.
   - Using point-slope form with point A(4, 2):
     $ y - 2 = -2(x - 4) $ 
     $ y = -2x + 8 + 2 $
     $ y = -2x + 10 $

2. The perpendicular slope to AB is $ \frac{1}{2} $ (negative reciprocal).
   
3. Using point-slope form with point C(6, 8):
   $ y - 8 = \frac{1}{2}(x - 6) $
   $ y = \frac{1}{2}x - 3 + 8 $
   $ y = \frac{1}{2}x + 5 $

So, the equation of the altitude passing through corner C is $ y = \frac{1}{2}x + 5 $.

To find the endpoints of the altitude, we set $ y = \frac{1}{2}x + 5 $ equal to the equation of side AB, $ y = -2x + 10 $, and solve for x and y:

$ -2x + 10 = \frac{1}{2}x + 5 $
$ -\frac{5}{2}x = -5 $
$ x = 2 $

Plugging x = 2 back into either equation gives us y = 6.

So, the endpoint of the altitude is (2, 6). The length of the altitude is the distance between C(6, 8) and (2, 6), which can be calculated using the distance formula:

$ \text{Distance} = \sqrt{(6 - 2)^2 + (8 - 6)^2} $
$ = \sqrt{4^2 + 2^2} $
$ = \sqrt{16 + 4} $
$ = \sqrt{20} $
$ = 2\sqrt{5} $.

Therefore, the endpoints of the altitude are (2, 6) and (6, 8), and the length of the altitude is $ 2\sqrt{5} $.